Hobbing and Grinding Process Modelling of Helical

Wheel in The Speed Reducer of an Electric Vehicle

Master's Thesis in Mechanical Engineering

Supervisors: Jérôme BRUYERE François GIRARDIN

Tutor: Francesca Maria CURÀ

Grad student: Nagelaa ES SAFIANI

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3

MASTER'S THESIS IN MECHANICAL ENGINEERING

Hobbing and Grinding Process Modelling of

Helical Wheel in The Speed Reducer of an

Electric Vehicle

Nagelaa ES SAFIANI

Department of Mechanical Engineering

Division of Industrialization and Processes National Institute of Applied Sciences of Lyon

Lyon, France 2019

Department of Mechanical Engineering

Division of Production Polytechnic University of Turin

Turin, Italy 2019

4

Abstract

When developing a mechanical transmission one important characteristic of the

transmission is how much tonal noise and vibrations are generated. The

vibrations causing are called gear whine noise and they are generated in the gear

contacts of the transmission and propagate through the shafts and bearings to

the housing, where they become airborne. In electrical vehicle applications the

absence of a loud combustion engine makes the gear whine noise more distinct

and easily perceived by the human ear. The main cause of the noise has been

assumed to be excitations due to variations of tooth profile errors in the wheel of

speed reducer. For this reason, a code to simulate the hobbing and grinding

process, that generates the helical wheel, has been developed by the MATLAB

software. To write the code, a kinematic study about hobbing and grinding

process has been performed, then it has been possible to obtain the theoretical

points of contact on wheel surface and hob surface. After obtaining the

programming code, it will be possible to compare the theoretical and real

surfaces and analyse the possible causes.

5

Résumé

Pendant le développement d’une transmission mécanique il est important de

maitriser le bruit et les vibrations générées. Le spectre de ces signaux peut faire

apparaitre certaines fréquences indépendantes de la fréquence d’engrènement.

Elles sont appelées raies fantômes et sont générées par le contact entre les roues

d’une transmission. Elles sont transmises à travers les arbres et les roulements

vers le carter. Dans un véhicule électrique, l’absence de moteur thermique fait

que le bruit est plus perceptible pour les personnes. Les erreurs de fabrication

de profil des roues sont considérées comme la causes principales de ce bruit. Pour

cette raison et, par le moyen d’un code MATLAB, le processus de taillage et le

processus de rectification ont été simulés. Pour le développement du code, on a

étudié la cinématique de taillage et de rectification pour obtenir les points de

contact théorique sur la surface de la roue et sur la surface de la fraise. Par la

suite, il sera possible de comparer les surfaces théoriques et les surfaces réelles

de la roue pour essayer de comprendre les causes du bruit.

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Acknowledgements

I would like to thank my supervisors Francois Girardin and Jerome Bruyere for

their great help and guidance throughout this thesis, for the continuous support

of my Master study and related research, for their patience, motivation, and

knowledge. Many thanks to the staff of Groupe PSA for their time and sharing

of knowledge. I would also like to give thanks to my referent Francesca Maria

Curà for taking care of me during my academic mobility.

I must express my very profound gratitude to my parents and my sister for

providing me with unfailing support and continuous encouragement throughout

my years of study and through the process of researching and writing this thesis.

A special gratitude to Imad for his patience and help, this accomplishment would

not have been possible without him.

Finally, I would like to thank God to have given me a life full of love.

Nagelaa Es Safiani

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Contents

1 Introduction ................................................................................................................ 13

1.1 Objetcives ............................................................................................................ 14

2 Review of Literature ................................................................................................ 15

2.1 Mechanic Transmission ....................................................................................... 15

2.1.1 Transmission in Automotive ........................................................................ 17

2.2 Speed Reducer and Gearbox ............................................................................... 22

2.3 Manufacturing Process ......................................................................................... 22

2.3.1 Machining ........................................................................................................ 22

2.3.2 Heat Treatment .............................................................................................. 23

2.3.3 Gear Wheel Production Line........................................................................ 24

3 Manufacturing Process .......................................................................................... 26

3.1 Hobbing .................................................................................................................... 26

3.1.1 Manufacturing defects .................................................................................. 31

4 Gear Whine Noise ...................................................................................................... 39

4.1 Description of gear profile measuring machine .............................................. 41

4.2.1 Profile undulations......................................................................................... 45

4.2.2 Helix undulations ........................................................................................... 46

4.2.3 Generation of undulations............................................................................ 48

5 Method ........................................................................................................................... 51

5.1 programming software ......................................................................................... 52

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5.1.1 Matlab ............................................................................................................... 52

5.2 Modeling of the cutting process. ......................................................................... 57

5.2.1 Helical pinion reference system .................................................................. 57

5.2.2 Modeling of the hobbing kinematics .......................................................... 62

5.2.3 Technical data for pinion and cutter .......................................................... 66

6 Results ........................................................................................................................... 71

6.1 Helical pinion drawing ......................................................................................... 71

6.2 Hob drawing ............................................................................................................ 73

6.3 Pinion-Hob engagement ....................................................................................... 74

6.4 Contact points during grinding process............................................................ 77

6.5 Contact points during the hobbing process ..................................................... 84

7 Conclusion .................................................................................................................... 87

7.1 Project development .............................................................................................. 87

7.2 Forthcoming objectives ......................................................................................... 88

Bibliography ................................................................................................................... 90

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Nomenclature αn real pression angle

αt apparent pression angle

a tip level

A amplitude angular deviation

β helix angle

βa feed motion angle

βb base helix angle

b tooth width

C interaxial distance

Ca addendum modification coefficient

Cf dedendum modification coefficient

Cf profile control

Cfu effective dedendum modification coefficient

D pitch diameter

da tip diameter

db base diameter

df root diameter

ԑ defect

e eccentricity

φ cylindrical coordinate

Fa tip form

Ffα profile form deviation

Ffβ helix form deviation

fh hob feed motion per turn

fHα profile slope deviation

fHβ helix slope deviation

Fα total profile deviation

Fβ total helix deviation

gα length of path of contact

i tooth number

k flute number

λβ wavelength undulation

Lα profile evaluation length

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Lβ helix evaluation length

mn real module

mt apparent module

na feed motion direction

Nbg number of hob flutes

Nf start of active profile

Ωh hob angular speed

Ωp pinion angular speed

p circle pitch

ρ coefficient of tooth side

r pitch radius

ra tip radius

rb base radius

rf root radius

Rh hob mobile referent system

Rh0 hob fix referent system

Rp pinion mobile referent system

Rp0 pinon fixe referent system

S angle between rotation axes of pinion and hob

s linear extension of the teeth on the action plane

sn real thickness

st apparent thickness

stb base apparent thickness

τԑ defect torsor

Θh hob angular position

Θp pinion angular position

Vh feed motion speed

x0 eccentricity error

x1 pinion profile shift coefficient

x2 hob profile shift coefficient

Zeq speed ratio

zh number of hob fillets

zp number of pinon teeth

12

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1

Introduction

The project is carried out for the Groupe PSA company known Peugeot S.A. Is a

French multinational manufacturer of automobiles and motorcycles. The object

of study is the helical tooth in the gearbox of an electric vehicle. The

transmission of a gear normally presents noises that depend on the meshing

frequency between the gear teeth and its harmonics, then they are remark

frequencies. This noise phenomenon can be controlled since the theoretical

geometric definition of the helical tooth. Noise from electric vehicles is an

important environmental issue and several sources contribute to the tonal noise

level. The tonal noise from the gearbox can be very disturbing for the driver,

even if the noise level from the gearbox is lower than the total noise level. The

human ear has a remarkable way of detecting pure tones of which the noise from

loaded gears. The gearbox can be an important contributor to the total noise

level1. Some tonal noises can occur, which are not directly linked to the geometric

definition of the helical wheel, one hypothesis is that the problem is linked to the

surface texture. With surface texture we mean the post-treatment processes to

reinforce the component and all manufacturing process to have the tooth profile.

In order to better control tonal sources which could eventually lead to

undesirably high levels of gear whine, one need is to better understand and if

14

possible, to model all those manufacturing processes that affect the mechanical

performance of the component. From the cutting process, to the grinding process

to thermal-treatment processes.

1.1 Objetcives

In order to proceed with the analysis of the mechanical problem therefore

necessary to study the kinematics of the machines used to manufacturing the

helical gear, the kinematics of the cutting machine, the kinematics of the

grinding machine and the others possible manufacturing processes of the

component. By studying the kinematics of manufacturing machines, it is possible

to achieving the nominal geometry of the component and then to possible

geometrical and surface finishing problems. After analyzing the nominal

geometry of the helical tooth, different errors will be introduced to observe the

behavior of the component in the transmission. We will analyze errors

concerning the surface finishing and errors related to assembly. Will be done a

comparison between the different surfaces finishing related to various errors

introduced and the respective results.

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2

Review of Literature

This chapter explains what a mechanical transmission is and how it is used in

the automotive sector. In the automotive sector there are different types of

mechanical transmissions, therefore the most common in the industrial field will

be explained.

2.1 Mechanic Transmission

A transmission is a machine in a power transmission system, which provides

controlled application of the power. Often the term transmission refers simply to

the gearbox that uses gears and gear trains to provide speed and torque

conversions from a rotating power source to another device2. The term

transmission indicates all the necessary components for starting and moving a

vehicle and mechanical machine. The main components that make up a

transmission are : the transmission shaft, the gearbox, the clutch, the

differential. A generic representation of gearbox of vehicle and its components,

are shown in Fig.2.1.

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Fig.2.1 Mechanic Transmission3

The transmissions are widely used in motor vehicles where the use of the

transmission has the objective of adapting the engine power to the torque of the

driving wheels. The engines of the vehicles are characterized by a high rotation

speed but having a high rotation speed is not appropriate for starting and

stopping. The role of a transmission is mainly to reduce the speed provided by

the engine to the speed required by the driving wheels. The transmission is

characterized by the transmission ratios that allow to know the ratio in the

variation of the speeds. There are transmissions that change over manually or

automatically, there are single or multiple ratio transmissions. There are also

hybrid configurations.

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2.1.1 Transmission in Automotive

The need for a transmission in an automobile is a consequence of the

characteristics of the internal combustion engine. Engines typically operate over

a range of 600 to about 7000 rpm, while the car's wheels rotate between 0 rpm

and around 1800 rpm4. The main shaft extends outside the case in both

directions: the input shaft towards the engine, and the output shaft towards the

rear axle. The main bearings support the shaft while a pilot bearing, at the point

of split, holds the other shafts together. The gears and clutches ride on the main

shaft, the gears being free to turn relative to the main shaft except, when

engaged by the clutches.

The main vehicle transmission types are:

• Manual

• Semi-Automatic

• Automatic

2.1.1.1 Manual Transmissions

There are two principal kinds of manual transmission, the synchronized one and

unsynchronized one. The unsynchronized transmissions need a manual

synchronizing. The skill of the driver is important for a manual synchronizing

where at each shift event he need to synchronize gear speeds. This kind of

transmission is usually used in motorsport applications and heavy commercial

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vehicles. Most modern manual-transmission vehicles are fitted with a

synchronized gear box, as shown in Fig.2.2.

Fig.2.2 Manual Transmission5

Transmission gears are always in mesh and rotating, but gears on the principal

shaft can freely rotate or be locked to the shaft. To lock the gears to the principal

shaft it is necessary to use a collar on the shaft. This collar can slide sideways so

that teeth on its inner surface bridge two circular rings with teeth on their outer

circumference: one attached to the gear, one to the shaft hub. When the rings

are bridged by the collar, the gear is rotationally locked to the shaft and

determines the output speed of the transmission. The gearshift lever manages

the collars and one collar may be permitted to lock only one gear at any one time.

The operators must understand how to shift the transmission into and out of

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gear. A non-synchronous transmission is a form of transmission based on gears

that do not use synchronizing mechanisms, then this transmission requires an

understanding of gear range, torque, engine power, range selector, multi-

functional clutch, and shifter functions.

2.1.1.2 Semi-Automatic Transmission

A semi-automatic transmission is a manual transmission with little actuators

that do all the physical stuff that the driver would ordinarily be doing, as shown

in Fig.2.3.

Fig.2.3 Semi-Automatic Transmission6

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Often the transmission itself is a manual transmission that has had the

additional semi-automatic components fitted to it. Actuators attached to the

outside of the transmission take the place of a gear lever, selecting the

appropriate gear for the driver. As the transmission itself is often just a standard

manual gearbox, there will generally be two actuators to simulate both the up

and down motions of a gear lever7. The clutch is controlled by a hydraulic pump,

this pump simulates the pressing of the driver on the clutch pedal. All this

system is managed by a control module that decides when to change gears. It is

known that the semi-automatics have a better driving comfort compared to an

automatic one, it also presenting many characteristic advantages of the manual

ones. But the gear changes of the semi-automatic do not show a fluidity present

instead in the automatics.

2.1.1.3 Automatic Transmission

An automatic transmission is characterized by the possibility of being able to

automatically change the gear ratio of the vehicle in motion without the manual

use of the driver. This type of transmission is mainly used in internal combustion

vehicles characterized by a high speed of rotation and by important torque.. The

most popular form found in automobiles is the hydraulic automatic

transmission, as shown in Fig.2.4.

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Fig.2.4 Automatic Transmission8

The characteristic of this system is using a fluid coupling and not using the

system characterized by a friction clutch. Gear changes are also managed thanks

to the action of locking and unlocking a planetary gear system in a hydraulic

way. These systems are also defined by different gear ranges and are

characterized by a safety system that can be defined as a parking pawl that

blocks the transmission output shaft to prevent the vehicle from advancing or

dipping. In the case of machines with limited speed ranges or fixed engine speeds

they simply use a torque converter to provide variable engine transmission to

the wheels.

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2.2 Speed Reducer and Gearbox

The mechanical problem that is treated in the project concerns a gearbox for a

thermal engine. The problems that are present in the gearbox, that is already in

use and installed in vehicles, are to be avoided in the speed reducers that are

used in electrfied vehicles.

2.3 Manufacturing Process

Generally, a mechanical component is characterized not only by its function in a

mechanical system, but also by its material with which has been manufactured,

by the mechanical manufacturing and heat treatment to which it has been

subjected.

2.3.1 Machining

The main components of the gearbox are made of aluminum, steel or cast iron. A

series of machining operations, performed with numerically controlled

machines, allow to give them their final shape: turning, cutting, shaving, heat

treatment, phosphating, grinding, lapping. The quality of the components is

measured by means of control integrated into the machines Fig.2.7 and by

statistical samples taken by the operators.

To ensure that te gearboxes operate properly, the teeth of the pinions and shafts

must be adjusted to the nearest micron. To make micron toothings, the

machining tools are sharpened and measured with maximum precision.

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Fig.2.7 Numerical Control Machine9

2.3.2 Heat Treatment

After machining, all parts, except casings and housings, are subjected to heat

treatment for carbonitriding operations. Carbonitriding is a

metallurgical surface modification technique that is used to increase the

surface hardness of a metal, thereby reducing wear. During the process, atoms

of carbon and nitrogen diffuse interstitially into the metal, creating barriers

to slip, increasing the hardness and modulus near the surface. Carbonitriding is

often applied to inexpensive, easily machined low carbon steel to impart the

surface properties of more expensive and difficult to work grades of steel.

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Made in powerful furnaces Fig.2.8, this operation consists of treating the parts

by heating them to 860 ° C to make them more resistant to mechanical stress

and wear. This operation is essential to guarantee the durability of the gearbox.

Fig.2.8 Heat Treatment Furnace9

2.3.3 Gear Wheel Line Production

To produce the gear wheels, the cutting process is carried out through the

hobbing process, and the grinding process using an abrasive wheel. Not all

wheels are subjected to the same production process.

In order to study the harmonic phenomenon and understand its nature, the

production cycle is as follows:

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• Production of three pieces subjected to hobbing process, after heat

treatment and finally to the grinding process.

• Production of a piece subjected to hobbing process, after shaving, heat

treatment and finally to the grinding process.

It can therefore be observed that the difference between the two types of

components produced is the presence or absence of the shaving process before

heat treatment. It is important to note that the four gear wheels are produced

on the same machine. It is necessary to study the production processes and

understand if errors occur after the hobbing process or after the shaving or

grinding process. Understand if the error is related to the machine or to other

factors and searching for possible solutions. In order to carry out a detailed study

of the possible causes, it is necessary to study the kinematic that manages the

movement of the numerical control machines used in the production of helical

wheel.

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3

Manufacturing

Process

Most modern gears are cut by a process known as generating cutting. During the

cutting process the cutter and the gear blank are each rotated, as if they were a

meshing gear pair. One of most common methods by which gears are generated

is hobbing.

3.1 Hobbing

The hobbing system is characterized by a transmission system that aims to vary

the speed and torque provided by the engine. The transmissions in this complex

are mainly gear transmission and belt drive. The system, which is characterized

by different shifting levels, allows to manage the speed of the machine according

to the manufacturing needs. The strict speed and itinerary synchronization

relationship of inline transmission chain is relied on transmission components

having accurate transmission ratio, and therefore gear itself manufacturing

error and assembly error on the accuracy of the final hob is the major role10.

27

The relative movement of hob and workpiece is based on linear movement and

rotational movement of each kinematic pair, including X linear motion, Z linear

motion, Y linear motion, A, B and C rotary motion, which with machine bed, hob,

piece together constitute hobbing physical structure Fig.3.1.

Fig.3.1 Hobbing Machine10

A typical hob Fig.3.2 has basically the same shape as a screw, with one or more

threads, and each thread is cut by a few gashes, either at right angles to the

thread, or parallel with the hob axis, so that cutting faces are formed.

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Fig.3.2 Hob11

When the hobbing system works, the hob is meshed to a work gear and during

the meshing, the hob creates the teeth surfaces of an imaginary rack, as shown

in Fig.3.3. It is necessary to have a enough profile of the rack tooth if we want to

have a good involute profile of the work gear. It is also neccesary to have a good

accuracy of the hob if we want a good result. It is important to consider also the

contribute of hob mounting. To analyse the hobbing process normally we use

concept of imaginary rack because the cutting process of the hob is similar to

that of a rack cutter. The teeth of a hob must represent a helical rack moves in

the direction of tangent to a pitch circle of a work gear with helix angle while the

hob is rotated. It is necessary to incline the hob arbor axis at angle of the helix

angle minus or plus the lead angle of the hob thread.

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Fig.3.3 Imaginary rack, Generating center, Number of cutting tooth12

The hobbing process has two advantages over that of a rack cutter: in both the

axial and the tangential directions the motions of the cutting faces are

continuous, so there is not necessary to move the hob back after that, one or two

teeth are cut (as in the rack cutting system) and the cutting action is obtained

without any reciprocating motion of the hob and the hob is fed slowly in the

direction parallel to the gear axis. As we said previously, each hob’s cutting face

simulates a tooth of the rack cutter, then the hob is tipped over by an angle called

swivel angle Fig.3.4, for meet the gear blank and the cutting faces at the same

angle of the rack cutter.

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Fig.3.4 Direction of hob feed13

The normal section is the one where the hob thread stays perpendicular to the

direction of the thread, and there are differents kind of hobs which are designed

to have the threads shaped in the normal section in the same way of the teeth of

a rack cutter. If we want to have the same shape of the cutting faces like the one

of the teeth's rack cutter, we need to have the gashes of the hob perpendicullary

to the direction of the threads. During the hobbing process, the rack cutting

system is simulated because the cutting pitch circle in the gear blank is the same

with its standrard pitch circle13. For the tooth thickness we need to pay attention

to two factors: the thickness of the hob and the depth to which the hob is fed into

the gear blank. During the hobbing process we cannot confirm that this process

is exactly like a rack cutting system because there are some errors, but these

errors are so small.

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3.1.1 Manufacturing defects

Although we plan a precise modeling of the machining processes, such as cutting

and grinding processes, defects are unavoidable. The main manufacturing

defects are due to errors related to assembly, to the kinematics of the cutting or

grinding machine and machine for making the tool. These defects therefore

induce disturbances on the theoretical cutting path. By analyzing the defects

present in the wheel tooth of a helical gear, the main origins of the defects are:

• Pinion assembly fault

• Defect of assembly of the cutting tool

• Operating fault of the cutting machine

• Operating fault of the grinding machine

Analytically we can define any manufacturing defect by characterizing it

through a displacement torsore. In general, we can call with ԑ the defect and

with τԑ the corresponding displacement torsore whose center is called Oԑ and

whose base is defined through the unit vectors (nԑx, nԑy, nԑz).

The displacement torsor is therefore defined as follows14:

𝜏𝜀 = 𝜀𝑅𝜀 = 𝑅𝑥. 𝜀𝑥 + 𝑅𝑦. 𝜀𝑦 + 𝑅𝑧. 휀𝑧

𝛿 𝜀𝑅𝜀(𝑂𝜀) = 𝑑𝑥. 𝜀𝑥 + 𝑑𝑦. 𝜀𝑦 + 𝑑𝑧. 휀𝑧

32

Considering then a generic point on the surface of the tooth and defined the

vector 𝑶𝜺𝑷𝑹𝜺 as the vector that joins the center Oԑ with the generic point P, we

can define the displacement of this point as14:

𝛿 𝜀𝑅𝜀(𝑃) = 𝛿 𝜀

𝑅𝜀(𝑂𝜀) + 𝜀𝑅𝜀⋀ 𝑝 𝑂𝜀𝑃

𝑅𝜀 (3.1)

When we have different manufacturing defects, we need to study and analyze

the combination of different effects. It is necessary to suppose the geometric

defects created during the cutting process with dimensions much lower than the

dimensions of the tooth, we will consider it with orders of magnitude of the

micron. After analyzing any manufacturing defect, the resulting geometric

defect that will be obtained will be given by the algebraic sum of all the

individual defects determined for each manufacturing defect. Before doing this,

the various manufacturing defects must be modeled analytically and related to

the same reference system.

3.1.1.1 Defects of the inaccuracy of the cutting machine.

Mainly it is possible to divide the defects caused by the imprecision of the

hobbing machine as follows:

• The defects related to the geometry of the machine, to the static defects of

misalignment of the different axes between them.

• The defects related to the movement of the machine, then the kinematic

problems of the machine.

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To verify the absence of misalignment errors of the machine axes, first must

choose one of the axes as the reference axis respect to which the measurements

is made. It is usually recommended to choose as a reference axis, an axis that is

common for the different types of cutting machines, for example the axis of

rotation of the workpiece platform where the pinion to be cut is installed, and we

can refer to the standard NF-ISO 6545. The standard authorizes, for a 500 mm

machine advancement stroke, misalignment defects of 3e-5 rad, therefore

relatively low values. The same study must also be carried out for the rotation

axis of the machine, where the cutting tool is positioned. Also, for the hob it is

fundamental that its axis of rotation is oriented according to the theoretical

orientation. Generally, to mesure the misalignment of hob are used two sensors

on both sides Fig.3.5.

Fig.3.5 Axis Misalignment12

34

Another type of error very common in mechanical system, is the transmission

error. This error is the rotation delay between driving and driven gear caused by

the disturbances of inevitable factors such as elastic deformation,

manufacturing error, alignment error in assembly15. Defining θ1 and z1,

respectively the angular position and the number of teeth of the driving wheel

and θ2 and z2, respectively the angular position and the number of teeth of the

driven wheel, the transmission error is defined as follows16:

𝐸𝑑𝑇 = θ2 −𝑧1𝑧2 θ1 (3.2)

It is also possible to define a generic equation to determine the transmission

error caused by eccentricity of a pinion of a cinematic chain as follows14:

𝐸𝑑𝑇(𝑖) =𝑒(𝑖)

𝑟𝑏(𝑖 − 1) 𝑍𝑒𝑞(𝑖 − 1) 𝑠𝑖𝑛(𝜃𝑒𝑖 + 𝑍𝑒𝑞(𝑖) 𝜃𝑝) (3.3)

Where:

e(i) eccentricity of the pinion (i)

θei angular phase of the eccentricity according to the pinion reference (i)

rb(i-1) base radius of the pinion (i-1)

Zeq(i-1) speed ratio between the pinion (i-1) and the workpiece platform on

which it is installed

Zeq(i) speed ratio between the pinion (i) and the workpiece platform on

which it is installed

θp angular position of the workpiece platform

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3.1.1.2 Defects caused by inaccurate assembly of the pinion and cutting tool

The assembly of the pinion and cutting tool can most often present deviations

from the theoretical assembly. The inaccuracy of the assembly can be given by

the linear combination of one or more deviations simultaneously. Taking as an

example the axis of rotation of the pinion: it can have an eccentricity, so it

remains parallel to the theoretical axis, but it is shifted. The same axis, now

shifted, can also have an angular deviation. The same problem can be found in

the assembly of the tool on the cutting machine, even for the tool rotation axis

we can have the displacment of the theoretical axis and an oscillation. For both

cases it is possible to calculate the error through a torsore that considers both

phenomenons, according to the local reference system of the pinion or hob.

Fig.3.6 shows the presence of eccentricity represented by the displacements Δx,

Δy, Δz and the angular deviations represented by the oscillations Δθx, Δθy, Δθz,

of a generic cylindrical structure, which can rappresents the pinion or the hob.

Fig.3.6 Linear and Aangular Deviations10

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3.1.1.3 Defects generated by the inaccurate grinding of the cutting tool

Grinding is an abrasive machining process that uses a grinding wheel as tool.

The grinding is a true metal-cutting process. Each grain of abrasive functions as

a microscopic single point cutting edge. Grinding is used to finish workpieces

that must show high surface quality and high accuracy of shape and dimension.

As the accuracy in dimensions in grinding is of the order of 0.000025 mm, in

most applications it tends to be a finishing operation and removes comparatively

little metal, about 0.25 to 0.50 mm depth17. However, there are some roughing

applications in which grinding removes high volumes of metal quite rapidly. The

grinding operation can concern both the hob and the wheel.

The tool used to perform the grinding operation is the grinding wheel. This is an

expendable wheel used for various grinding and abrasive machining operations.

It is generally made from a matrix of coarse abrasive particles, they are before

pressed and then bonded together to form a solid, circular shape and various

profiles. Grinding wheels may also be made from a solid steel or aluminium disc

with particles bonded to the surface. One example among many others are shown

in Fig.3.8.

Fig.3.8 One example of Grinding Tools18

37

If the cutter grinding operation is not carried out correctly, this error will affect

the cutting operation that will be carried out subsequently, thus obtaining low

quality gears. The hob error that is present during the grinding processis called

hob rake error. The rake error is described as the tooth face tolerance relative to

the centerline of the hob which is measured from the full cutting depth to the

outer diameter of the hob tooth19. The tooth face of a hob with zero rake design

extends on a theoretical line through the center of the hob as shown in Fig.3.9.

Fig.3.9 Zero Rake Error19

When the hob is grinding, the original rake angle of the tooth must be

maintained whether it was designed with a positive, radial or negative rake

angle. If an excessive stock has been removed from the lower part of the hob

tooth face the depth of the tooth has increased and we have a positive rake error

Fig.3.10. If there is a decrease of pressure angle, this produces an incorrect

involute profile when generating a gear. The gear tooth with a plus condition

toward the tip is produced by a decreased of pressure angle. This type of

38

condition may cause gear noise and premature failure. If the grinding or hobbing

wheel removes excessive stock from the upper portion of the hob tooth face we

have then a negative rake error. This meaning that the depth of the hob tooth is

deceased, but the pressure angle has increased. Rake error is usually caused by

an incorrectly positioned of grinding or hobbing wheel due for example for an

incorrect mounting, or an incorrectly dressed wheel. If we want to to correct this

kind of error, it possible to repositioning the principal axe of grinding or hobbing

wheel on the hob grinder.

Fig3.10 Positive and Negative Rake Error19

39

4

Gear Whine

Noise

When developing a mechanical transmission one important characteristic of the

transmission is how much tonal noise and vibrations it generates. The vibrations

causing the gear whine noise is generated in the gear contacts of the

transmission and propagate through the shafts and bearings to the housing,

where they become airborne. In electrical vehicle the absence of a loud

combustion engine makes the gear whine noise more distinct and easily

perceived by the human ear. There are many types of different noises associated

with gears, but one of the more distinct noises is the gear whine noise. This noise

is emitted from gears that are in mesh and the sound is characterized as

vibrations with frequencies same as the gear mesh frequency and its multiples.

The noise exhibits a periodic behaviour and it is therefore perceived as a tonal

noise. It is therefore an important factor when reducing the noise level of

transmissions since the human ear is more sensitive to tonal noises, compared

to noises with more random characteristics. Because the acoustic pressure, due

40

to gear whine noise, is not necessarily proportional to the engine speed the

unexpected noise and therefore the undesirable acoustic phenomena due to the

power system could be high in any engine running condition20. For that reason,

gear whine noise represents an important problem to solve or to reduce or to

manage at least in an appropriate way.

Many factors influence the vibro-acoustic emissions, mainly they are:

• Transmission error

• Variation of meshing stiffness

• Dynamical forces of meshing

• Friction forces

• Detention of air and lubricant between the teeth

The mainly cause of generation of the noise in the mechanical transmission are

the forces. When we have an important variation of forces, this causes the

vibration of different components present in the transmission. These forces

generally change in amplitude, direction or position and this changment is

caused by surfacing errors. The vibrations are then transmitted to the shafts

through the bearings to the housing, when the housing is excited the noise is

propagated to the vehicle and the noise is heard from the driver. Vibrations are

also transferred through the housing mountings where they can excite other

external components, such as parts of the compartment in a vehicle.

41

The flow chart below shows the noise path from the source to the external

environment.

4.1 Description of gear profile measuring machine

To inspect the presence of surfacing profile errors on the gear wheel, a specific

machine is used to control and check cylindrical gears, pinion cutters and

shaving cutters, worms and worm wheels, hobs. Specifcly this machine wants to

research different types of deviations in dimension, location and position of

symmetrical pieces. Further applications include also some cams and camshaft

measurement and rotor measurement. Klingelnberg measuring centres today

are one of the most important measuring center of this kind of machines and

engineering applications, they represent one of most important brands in

different sectors: automotive, commercial vehicle industries, aerospace,

aeronautical engineering industries.

Torsional and bending gear wheel

vibrations

Excitations due to the meshing

Dynamical forces of meshing and

bearing

Interaction between the

bearing and sheel

Noise

Radiant surfaces

42

This allows the following measurement tasks to be fully automated in a single

setup:

• Gear measurement

• Optical measurement

• General coordinate measurement

• Form and position measurement

• Roughness measurement

• Contour measurement

Fig.4.2 shows the trajectory effected by the head of the measuring machine for

obtaining lead line, pitch, concentricity and profile the tool used for the helical

gear profile measurement.

Fig.4.2 Measurement21

43

For an excellent precision measuring centre, it is necessary to consider the

accuracy, reproducibility and durable rotary table. Firstly, it is necessary to

configure the machine and guarantee the concentricy seating of the workpiece.

If the concentricity is checked, with the combination of three linear measuring

axes (tangential, radial and vertical), the precision measuring centres is able to

trace and control the functional surfaces of gearing and general mechanical

components. This guarantees the maximum measuring ratio of accuracy and

reproducibility.

This type of measuring centres used to control the quality of the component

normally are characterized by a heavy-duty, stable beds and guide system. The

bearings and guides are backlash-free at the measuring axes, this kind of

property is the basis for the high basic mechanical technologic of measuring

centres. The integrated 3D tracer head enables both discrete-point tracing and

scanning and continuos measuring.

4.2 Influence of tooth surface undulation on noise

To ensure vibration quality and quiet operation, gears are manufactured so that

their geometry is as close as possible to their design. In practice, however, there

are always irregularities during the manufacturing process that lead to various

kinds of gear errors and resultant vibration and noise problems. Usually gear

vibration is mainly observed at the fundamental tooth meshing frequency and

its harmonics. However, it is known that when gear errors exist, many other

44

frequency components are also observed at the same time. Common errors of

every gear tooth lead to vibration at meshing frequencies as well as the variation

of tooth stiffness during meshing. Pitch error and the other fluctuations from

common tooth shape bring about vibrations at low frequency bands and

sidebands surrounding meshing components22.

Vibration attributed to gear local defects such as crack or nick is observed as the

occurrence of an impulsive wave in time domain or a lot of comb spectra in wide

frequency range23.

There are also different kinds of errors that cause vibrations at the one order of

frequency that is not part of nominal of meshing frequency. It is so difficult to

find the real reason of this kind of vibrations because the frequency of these

vibrations is not related directly to some specified geometries or form of the gear.

The vibration source cannot be detected by ordinary tooth profile measurement

equipment, and for this reason, this vibration component is usually called ghost

noise. It is known that ghost noise is created by a periodic cyclic undulation on

the surface of a gear tooth continuing from one tooth to the next. The source of

this undulation can be traced to the irregularity in the process of generation.

The two types of undulation present during the manufacturing process are:

• Profile undulation

• Helix undulation

45

4.2.1 Profile undulations

The profile undulations are caused by errors present in the real profile of the

wheel, then the real profile of the wheel is different from the theoretical profile.

The profile undulation can be generated during the manufacturing process or by

wear and deterioration of the wheel during the gear meshing. ISO 1328 defines

the profile error as the amount by which a measured profile deviates from the

design profile. We have mainly three different kind of profile error: total profile

deviation, profile form deviation and profile slope deviation.

• Total profile deviation Fα

Distance between two facsimiles of the design profile which enclose

the measured profile over the profile evaluation range. The facsimiles of the

design profile are kept parallel to the design profile.

• Profile form deviation ffα

Distance between two facsimiles of the mean profile line which enclose

the measured profile over the profile evaluation range. The facsimiles of the

mean profile line are kept parallel to the mean profile line.

• Profile slope deviation fHα

Distance between two facsimiles of the design profile which intersect the

extrapolated mean profile line at the profile control diameter and the tip

diameter.

The mean profile line represents the shape of the design profile aligned with the

measured trace over the profile evaluation range. The profile evaluation range

46

represents the section of the measured profile starting at the profile control

diameter and ending at 95 % of the length to the tip form diameter24.

4.2.2 Helix undulations

The helix undulations are defined by ISO 1328 as amount by which a measured

helix deviates from the design helix. The helice deviation are measured in the

direction of base cylinder tangents, in the apparent plane.

We have mainly three different kind of helix error: total helix deviation, helix

form deviation and helix slope deviation.

• Total helix deviation Fβ

Distance between two facsimiles of the design helix which enclose

the measured helix over the helix evaluation range. The facsimiles of the

design helix are kept parallel to the design helix.

• Helix form deviation ffβ

Distance between two facsimiles of the mean helix line which enclose

the measured helix over the helix evaluation range. The facsimiles of the

mean helix line are kept parallel to the mean helix line.

• Helix slope deviation fHβ

Distance between two facsimiles of the design helix which intersect the

extrapolated mean helix line at the end points of the facewidth, b

The mean helix line represents the shape of the design helix aligned with the

measured. The helix evaluation range is the flank area between the end faces or,

47

if present, the start of end chamfers, rounds, or other modification intended to

exclude that portion of the tooth from engagement, that is, unless otherwise

specified, shortened in the axial direction at each end by the smaller of 5 % of

the facewidth or the length equal to one module24.

Fig.4.4 shows profile and helix deviations with unmodified involute.

Fig.4.4 Profile and Helix modification24

48

Where:

Fa tip form

a tip

Cf profile control

Nf start of active profile

Lα profile evaluation length

gα length of path of contact

Lβ helix evaluation length

b facewidth

4.2.3 Generation of undulations

To understand the generation of undulations, the kinematic relation in the

grinding machine MAAG it has been treated in the report of Georges Henriot25,

he associated the origin of undulation to the workout platform from an error of

the worm wheel. This error is therefore generated by the rotation frequency of

the screw. One of most common error related to the screw is the eccentricity

measured profile

facsimile of design profile

mean profile line

facsimile of mean profile line

49

error. This causes an angular deviation of the screw respect to its theoretical

position. This angular deviation is a sine wave with amplitude A Fig.4.5.

𝐴 = 𝑥0𝑡𝑔(𝛼0) (4.1)

Where:

X0 eccentricity error per turn

α0 pression angle

Fig.4.5 Eccentricity error of worm wheel25

The tool or grinding wheel removes too much material or less if this error is

present, then the helical wheel has some ripples on the profile.

It is possible to represent the section of the teeth as an undulation with

wavelength 𝜆𝛽.

𝜆𝛽 =𝜋 𝑑

𝑍0𝑡𝑔(𝛽) (4.2)

Where:

d primitive diameter

Z0 number of teeth

β helix angle

50

When the wheel moves it removes the metal following some undulations having

the angle of inclination of the generators δ Fig.4.6.

𝑡𝑔(𝛿) = 𝑡𝑔(𝛽)𝑠𝑖𝑛(𝛼𝑛) (4.3)

Fig.4.6 Undulation25

If we consider the meshing between the wheel and the pinion, the meshing meets

different ripples during the work. This generates an angular deviation of the

wheel and during the meshing there will be a noise which frequency will be:

𝑛 𝑍060

𝐻𝑧

Where:

n turn ∙ min-1 of the wheel

51

5

Method

In the previous chapters, it was highlighted that in mechanical transmission

system, gear pairs are widely used in automotive industries because of its

accurate transmission ratio and good reliability. Noise is created during the

engagement of gear pairs system and the vibration comes down to excitation of

gear transmission error. Many literatures have revealed that transmission error

is the main parameter that affects the performance and quality of transmission

of gear pairs, especially for noise vibration harshness performance of vehicle.

We decided to introduce some profile modification of the gear pairs to try to

detect the reason and the source of the gear whine noise. This kind of method is

already adopted by other researchers to study this phenomenon.

For exemple, modifying the tooth profile it is possible to decrease the interference

during the meshing and compensate teeth deflection under load without

sacrificing the tooth strength. Another type of error that can be decreased by

tooth modification is the transmission error.

52

5.1 programming software

Before to proceed with kinematic study of manufacturing machines, it is

necessary to analyze different possible ways for project development. It is

possible to adopt two different ways of resolution: MATLAB or CATIA V5. Using

programming codes to design gear models is much simpler compared to

professional drawing software (Catia V5, SolidWorks, AutoCAD), especially for

designing tooth profile modification. If we want to proceed by first way, it is

necessary to begin with theoretical kinematics study of manufacturing machine,

analyzing different equations that establish the working trajectories of the

machine, then implemented these equations on MATLAB numerical calculation

program to generate the wheel surface. The second way is to generate the helical

wheel using CATIA V5 software. It is therefore necessary to analyze the

advantages and disadvantages of two software and then choose which one is best

to develop the project. In this section we will describe the characteristics of these

methods to solve the problem.

5.1.1 Matlab MATLAB is the acronym of matrix laboratory. It is a

multi-paradigm numerical computing environment and

proprietary programming language developed by

MathWorks. MATLAB allows matrix manipulations,

plotting of functions and data, implementation of

algorithms, creation of user interfaces, and interfacing

with programs written in other languages, for example:

53

C, C++, C#, Java, Fortran and Python26. MATLAB comes with feature that

allows you to design a graphic user interface, which can provide a convenient

way to the users who can interact with the electronic device without knowing

the programming code, but this is possible if a mechanical design is not required.

In our case, we want to design an engagement between a hob and a helical wheel.

To design gear models in MATLAB, the first step is to find out the functions and

the relationships of all the curves of the gear from mathematic and geometric

point of view, and then use proper MATLAB codes and functions to program it.

To draw the gear models by MATLAB codes is difficult at the beginning. Firstly,

all the functions and relationships of gear curves should be found out from

mathematic and geometric point of view. Then, we must to study all equations

to design the gear and the relationship between the gear and the hob during the

engagement and before it. Secondly, all the curves of gear models are made up

of a series of nodes in matrix form, so how to handle with matrix operation like

matrix addition, matrix rotation should be learnt. Thirdly, proper MATLAB

programming languages and functions should be used. MATLAB Graphic User

Interface is designed at last, which gathers all the programming codes in a single

file. In the Graphic User Interface, we can obtain different kinds of gear models

and draw then adding some tooth profile modifications. If we want we can also

study the helical wheel with CATIA using the design of the wheel made on

MATLAB, because it is possible to import the MATLAB data point file in CATIA,

then it is possible to open the modelling the wheel by CATIA functions.

54

5.1.2 CATIA V5

CATIA is the acronym of computer-aided three-

dimensional interactive application. It is a multi-

platform software suite for computer-aided design

(CAD), computer-aided manufacturing (CAM),

computer-aided engineering (CAE), PLM and 3D, developed by the French

company Dassault Systèmes. CATIA V5 is a complex engineering design

software used, among many others, in parametric 3D modeling, surface

modeling and cinematic and structural simulations. Commonly referred to as a

3D Product Lifecycle Management software suite, CATIA supports multiple

stages of product development (CAx), including conceptualization, design (CAD),

engineering (CAE) and manufacturing (CAM). CATIA facilitates collaborative

engineering across disciplines around its 3DEXPERIENCE platform, including

surfacing & shape design, electrical, fluid and electronic systems design,

mechanical engineering and systems engineering27.

CATIA facilitates the design of electronic, electrical, and distributed systems

such as fluid and heating, ventilation, and air conditioning systems, all the way

to the production of documentation for manufacturing. It contains

comprehensive libraries for standard parts and has the possibility to create user

defined libraries for repetitive content. Helical gears are parametric parts and

the geometry of a helical gear is controlled by known parameters. In CATIA from

specific geometric data on helical gears, the source solid is created as base for

the digital 3D gears family. There are many methods available for developing

55

profiles of gear and spline teeth. Most of the techniques are inaccurate because

they use only an approximation of the involute curve profile. In addition, the

involute curve by equation technique allows using either Cartesian in terms of

X, Y, and Z or cylindrical coordinate systems to create the involute curve profile.

Since gear geometry is controlled by a few basic parameters, a generic gear can

be designed by three common parameters namely the pressure angle α, the

module m, and the number of teeth Z. CATIA V5 uses these parameters, in

combination with its features to generate the geometry of the helical gear and

all essential information to create the model28.

Gear development using parametric method means that the dimensions control

the shape and size of the gear. The gear should have the ability to be modified,

and yet remain in a way that retains the original design intent. By storing the

relationships between the various features of the design and treating these

relationships like mathematical equations, it allows any element of the model to

be changed automatically and regenerate the new model. If we have a parametric

model, we can have a degree of design flexible and we can also allow curved

surfaces to be rationalized and building components of highly complex.

Geometric forms so they can be built economically and efficiently. Since models

are developed from the template the time consumed for gear modeling each time

based on the parameters assigned is reduced. In fact, few entries in the input

screen develops the gear in no time. Cost of development of the model is lesser

because of time. The human fatigue is also very less.

56

In the table below the advantages and disadvantages of MATLAB and CATIA

V5 are sown.

ADVANTAGES DISADVANTAGES

MATLAB - Accurate system

- creating specific

uncorrected and

corrected gear

profiles

- preparing the teeth

profile individually

- an expert user is not

required to use the

software

- Accurate study of

machine's

kinematics to create

the code

- long time to design

- laborious method

CATIA V5 - Study of machine's

kinematics is not

required

- no code is written

- low time to design

- inaccurate system

- creating ordinary

representation of

the gear

- parametric method

- expert user is

required to use the

software

The MATLAB software is chosen to design the helical gear. MATLAB is chosen

because it offers accurate results without approximations provided by the

57

software. Moreover, the accurate study carried out at the Polytechnic allowed

me to have the requisites to analyze the problem from a kinematic point of view

and the accurate knowledge of my supervisors about gearbox, manufacturing

process and their experience allowed us to decide MATLAB as a programming

software.

5.2 Modeling of the cutting process.

The working principle of the cutting process through the cutter It can be traced

to a gear between helical wheels. The tooth of the pinion is generated by the

teeth of the cutter, the gear is defined without clearance and the cutting takes

place simultaneously on the two sides of the pinion. To simulate the cutting

process, it is necessary to deepen the kinematics that regulate the relative

movements of the two components: pinion and cutter. After analysing the

relative position of the rotation axes of the pinion and the milling cutter, it is

possible to proceed to obtain the contact points during the meshing and fifth the

theoretical cutting points of the pinion tooth. Having obtained the theoretical

cutting points on the toothing of the pinion, a comparison between the theoretical

and real points can be made. Before proceeding with the modeling, it is necessary

to define the pinion geometrically, defining the reference system.

5.2.1 Helical pinion reference system

Two local reference systems have been defined for the two components,

respectively. The local reference system integral with the pinion during the

58

cutting process is called Rp Fig.5.1. The theoretical rotation axis, therefore,

without errors, is Zp. The origin of the reference system is given by the

intersection of the Zp axis and the lower face of the pinion perpendicular to this

axis. The Xp axis defines the symmetry axis of the first tooth of the toothing on

the plane whose Zp coordinate is zero. The third axis is Yp. The three axes are

oriented according to the unit vectors respectively nxp,nyp,nzp.

Fig.5.1 Pinion reference system14

The numbering of the teeth is performed in ascending order from the Xp axis to

the Yp axis Fig.5.2.

59

Fig.5.2 Numbering of pinion teeth14

To define a point belonging to the helical toothing, the apparent plane of the

tooth is taken as the reference plane. The apparent plane is defined by the Xp

and Yp axes. The apparent plane constantly has the perpendicular Zp axis. It is

possible to draw a tooth thanks to the involute of the basic circle of the toothed

wheel. Assuming that point A is on the tooth -thes i, a coordinate zm of the Zp

axis and belongs to a cylinder of radius rm whose axis is Zp, this point is defined

by three cylindrical coordinates: A(rA,θiA,zA). Fig.5.3 shows the apparent plane

of the helical tooth and its goniometric references.

60

Fig.5.3 Aapparent plane of tooth

The angular coordinate θiA is defined as follows:

θ𝑖𝐴 = θ𝐴 + θ𝑖𝑍 (5.1)

Where:

rb base radius

r pitch radius

ra circumference radius to which point A belongs

e(r) apparent thickness of the tooth at the pitch radius

e(ra) apparent thickness of the tooth at radius ra

61

Through this system of equations:

θ𝐴 = θ + φ − φ𝐴

𝜃 =𝑒(𝑟)

2𝑟𝜑 = 𝑖𝑛𝑣(𝛼𝑡)

𝜑𝐴 = 𝑖𝑛𝑣(𝛼𝑡𝐴)

𝑒(𝑟) =𝜋𝑚𝑡

2+ 2𝑥1𝑚𝑛𝑡𝑎𝑛(𝛼𝑡)

It is possible to express θiA as follows:

θ𝑖𝐴 = (𝑖 − 1) ∙2𝜋

𝑍+𝑡𝑎𝑛(𝛽𝑏)

𝑟𝑏∙ 𝑧𝐴 + 𝜌 ∙ (

1

2𝑟(𝜋𝑚𝑡

2+ 2𝑥1𝑚𝑛𝑡𝑎𝑛(𝛼𝑡)) + 𝑖𝑛𝑣(𝛼𝑡) − 𝑖𝑛𝑣(𝛼𝑡𝐴)) (5.2)

Where:

𝛼𝑡𝐴 = 𝑎𝑐𝑜𝑠 (𝑟𝑏𝑟𝐴) (5.3)

Represents the apparent pressure angle of point A and depending on the side we

are considering ρ we can have the following values:

𝜌 = +1 𝑙𝑒𝑓𝑡 𝑠𝑖𝑑𝑒

−1 𝑟𝑖𝑔ℎ𝑡 𝑠𝑖𝑑𝑒

Defined the reference system of the helical pinion, it is possible to modeling the

cutting kinematics.

62

5.2.2 Modeling of the hobbing kinematics

During the hobbing process, the hob and pinion rotate around their respective

axes. The axes of rotation of the pinion and of the drill are not parallel, this

allows to obtain a helical pinion. During the process, the pinion only performs a

rotary movement around its own axis at a known angular velocity Ωp, the cutter

rotates about its axis at a known angular velocity Ωh and at the same time

translates to complete the pinion cutting.

The translation performed by the cutter takes place at a known speed Vh. We

have not spoken directly of axial translation because the translation can take

place in two different directions. We talk about the axial translation if the cutter

moves along the axis of the cut pinion, we talk about the oblique translation if

the hob moves along the direction of the toothing generated during the hobbing

process according to an angle of inclination β between the axis of rotation of the

pinion and the direction of advancement of the hob. Fig.5.4 shows the possible

feed directions of the hob.

63

Fig.5.4 Hob feed directions14

To carry on the modeling of hobbing and grinding process, we need to establish

the mathematical relationships between the pinion and the hob kinematics. We

have defined a fixed local reference system for the pinion Rp0 defined by nxp0,

nyp0, nzp0 and the origin Op. The fixed local reference system of the hob Rh0 is

defined by nxh0, nyh0, nzh0 and the origin Oh. Fig.5.5 shows the mutual position

of the reference systems Rp0 and Rh0.

64

Fig.5.5 Mutual position of pinion and hob14

The cutting motion direction is done by the unit vector na and it is defined as

follows:

𝒏𝑎 = 𝑐𝑜𝑠(𝛽𝑎)𝒏𝑧0 + 𝑠𝑖𝑛(𝛽𝑎)𝒏𝑦0

Depending on the type of feed βa will have the following values:

𝛽𝑎 = 𝛽 oblique cutting motion direction

𝛽𝑎 = 0 axial cutting motion direction

65

In modeling we assume that the feed is axial. The S coefficient represents the

algebraic sum of the primitive helical angle of pinion and hob.

By defining with θp and θh the angular position of the pinion and the hob,

respectively, during the hobbing process, it is possible to define a mobile local

reference system for the pinion and for the hob. The fixed local reference system

of the pinion Rp0 is expressed as a function of the fixed local reference system

Rh0 as follows:

𝒏𝑥𝑝0 = −𝒏𝑥ℎ0

𝒏𝑦𝑝0 = −𝑐𝑜𝑠(𝑆)𝒏𝑦ℎ0 + 𝑠𝑖𝑛(𝑆)𝒏𝑧ℎ0

𝒏𝑧𝑝0 = 𝑠𝑖𝑛(𝑆)𝒏𝑦ℎ0 + 𝑐𝑜𝑠(𝑆)𝒏𝑧ℎ0

The local mobile reference system of the pinion Rp is expressed as a function of

the fixed local reference system Rp0 as follows:

𝒏𝑥𝑝 = 𝑐𝑜𝑠(𝜃𝑝)𝒏𝑥𝑝0 − 𝑠𝑖𝑛(𝜃𝑝)𝒏𝑦𝑝0

𝒏𝑦𝑝 = 𝑠𝑖𝑛(𝜃𝑝)𝒏𝑥𝑝0 + 𝑐𝑜𝑠(𝜃𝑝)𝒏𝑦𝑝0

𝒏𝑧𝑝 = 𝒏𝑧𝑝0

The local mobile reference system of the hob Rh, is expressed as a function of

the fixed local reference system Rh0, as follows:

𝒏𝑥ℎ = 𝑐𝑜𝑠(𝜃ℎ)𝒏𝑥ℎ0 + 𝑠𝑖𝑛(𝜃ℎ)𝒏𝑦ℎ0

𝒏𝑦ℎ = −𝑠𝑖𝑛(𝜃ℎ)𝒏𝑥ℎ0 + 𝑐𝑜𝑠(𝜃ℎ)𝒏𝑦ℎ0

𝒏𝑧ℎ = 𝒏𝑧ℎ0

66

The angular positions θp and θh are not independent from each other so it is

possible to express θh as a function of θp as follows:

𝜃ℎ = −𝜋

𝑧2−𝑟1𝑟2∙𝑐𝑜𝑠(𝛽1)

𝑐𝑜𝑠(𝛽2)∙ 𝜃𝑝 (5.4)

Where r1 and r2 are the pitch radius of the pinion and the hob respectively.

5.2.3 Technical data for pinion and cutter

The pinion is generally defined by the following data:

- Number of teeth z1

- Real Module mn

- Real pressure angle αn

- Primitive helix angle β1

- Correction coefficient x1

The hob is generally defined by the following data:

- Number of flutes z2

- Number of gorge Nbg

- Real Module mn

- Real pressure angle αn

- Primitive helix angle β2

- Correction coefficient x2

67

The technical data provided by PSA show different values of real module and

real pressure angle between pinion and hob. The real module and the real

pressure angle between the pinion and the hob must be the same to allow the

engagement.

The pinion data are related to a local reference system, therefore a translation

of the technical data of the pinion in the reference system of the hob has been

carried out. The hob manages the hobbing process and therefore the teeth

created, for this reason the following considerations have been assumed:

a) Real pinion module equal to the real module of the hob

b) Real pinion pressure angle equal to the real pressure angle of the hob

Furthermore, the tip radius and root radius of the pinion, declared by PSA, have

been maintained the same, because they belong to a real geometry and they do

not depend on any reference system. Also, the base radius of the pinion, declared

on the data sheet, has been maintained the same to keep the same involute of

the helical tooth.

The Cfu coefficient has been assumed unitary. Starting from the module and the

real pressure angle, the pinion data have been recalculated. The data taken by

PSA data-sheet are represented without apex, the data recalculated are

represented with apex.

68

The mathematical relationships for data transformation are the following29:

𝛼𝑛′ = 𝛼𝑛ℎ

𝑡𝑎𝑛(𝛽𝑏)

𝑟𝑏=𝑡𝑎𝑛(𝛽𝑏′)

𝑟𝑏′

𝑟𝑏′ = 𝑟𝑏

𝑠𝑖𝑛(𝛽′) =

𝑠𝑖𝑛(𝛽𝑏′)

𝑐𝑜𝑠(𝛼𝑛′)

𝑚𝑛′ = 𝑚𝑛ℎ

𝑡𝑎𝑛(𝛼𝑡′) =

𝑡𝑎𝑛(𝛼𝑛′)

𝑐𝑜𝑠(𝛽′)

𝑟𝑎′ = 𝑟𝑎

𝑚𝑡′ =

𝑚𝑛′

𝑐𝑜𝑠(𝛽′)

𝑟𝑓′ = 𝑟𝑓

𝑟1′ =

𝑚𝑡′ 𝑧12

𝐶𝑎′ =

𝑟𝑎′ − 𝑟1′

𝑚𝑛′− 𝑥1

′

𝐶𝑓′ =

𝑟1′ − 𝑟𝑓′

𝑚𝑛′+ 𝑥1

′

𝑠𝑡′ =

𝜋 𝑚𝑡′

2+ 2𝑥1

′ 𝑚𝑛′ 𝑡𝑎𝑛(𝛼𝑡

′)

69

The coefficient x1’ forms part of the pinion thickness formula and, to calculate

the new value, several steps have been carried out as shown below:

a) Express the apparent base thickness, declared on PSA data sheet, in

according to the pitch radius

stb=st(rb) → st(r1)

𝑠𝑡 = (𝑠𝑡𝑏𝑟𝑏− 2𝑖𝑛𝑣(𝛼𝑡)) ∙ 𝑟1 (5.5)

b) Equal the thickness calculated with the general thickness formula as a function

of the recalculated pitch radius

st(r1’) → st’

𝜋 𝑚𝑡′

2+ 2𝑥1

′ 𝑚𝑛′ 𝑡𝑎𝑛(𝛼𝑡

′) = (𝑠𝑡2 𝑟1

+ 𝑖𝑛𝑣(𝛼𝑡) − 𝑖𝑛𝑣(𝛼𝑡′)) ∙ 2𝑟1

′ (5.6)

c) Obtain x1’

Through the PSA technical drawing of the hob, it has been possible to take the

geometric data of the hob. The profile shift coefficient has been calculated

starting from the thickness of the hob, and we obtained a null profile shift profile

or the hob. Now we can begin to model the engagement between pinion and hob.

70

The modelling consists in the following steps:

I. Draw the pinion profile

II. Draw the hob profile

III. Draw the engagement between pinion and hob, obtaining tangent profiles

IV. Get contact points on the wheel surface during the grinding process

V. Get contact points on the hob surface during the grinding process

VI. Get contact points on the wheel surface during the hobbing process

VII. Get contact points on the hob surface during the hobbing process

71

6

Results

In this section, the results about the modelling of hobbing and grinding process

will be shown and explained.

6.1 Helical pinion drawing

The equation (5.2) allows to trace the profile of the tooth on the apparent plane.

The vector rm1 defines the value of the radius cylinder on which calculate the

tooth profile. We introduced a factor dec to represent the desplacement to be

carried out to report the tooth profile on the origin of the reference system Rp0.

A null dec means tracing the tooth profile on a reference system that is shifteed

by half the thickness of the tooth. To return the tooth drawn on the origin of

reference system Rp0, a translation equal to dec is necessary.

After defining the profile of a pinion tooth on the Xp0 and Yp0 plane with Zp0

null, the matrices MX1, MY1, MZ1 have been created to define the 3D profile of

the pinion tooth. The vector tz defines the extension of the tooth on the Zp0 axis,

then its width. Fig.6.1 shows the tooth profile of the helical pinion.

72

Fig.6.1 Tooth profile of helical pinion

After defining the three-dimensional structure of the pinion tooth, the M2X1,

M2Y1, M2Z1 matrices have been defined to generate the three-dimensional

structure of the pinion. The following teeth are designed in according to a

circular pitch p. Fig.6.2 shows the pinion.

Fig.6.2 Helical pinion

73

6.2 Hob drawing

For the hob geometrical modelling, it is possible to consider the hob as a wheel.

Then, it is possible to consider the engagement between the hob and the pinion

as an engagement between two helical wheels. For these reasons, the process of

modelling the hob profile is the same used for the pinion previously. The rm2

vector defines the value of the radius cylinder on which calculate the tooth profile

of the hob.

After defining the fillet profile of the hob on the Xh0 and Yh0 plane with Zh0

null, the matrices MX2, MY2, MZ2 have been created to define the 3D profile of

the hob fillet. The vector tz2 has been defined to define the extension of the fillet

on the axis Zh0, then its width. Fig.6.3 shows the hob fillet.

Fig.6.3 Hob fillet

74

After defining the three-dimensional structure of the hob fillet, the M2X2,

M2Y2, M2Z2 matrices have been defined to generate the three-dimensional

structure of the hob. The following fillets are designed in according to a circular

pitch p. Fig.6.4 shows the hob.

Fig.6.4 Hob

6.3 Pinion-Hob engagement

The interaxial distance between pinion and hob is not given by the sum of their

pitch radius, but we must consider a distance given by the shift profile coefficient

of pinion and hob. This interaxial distance DC is shown in Fig.6.5.

75

Fig.6.5 Interaxial distances13

After defining the interaxial distances, the initial angular positions of the pinion

and the hob have been defined.

The initial angular position of the pinion is null and the initial angular is as

follows:

𝜃ℎ0 =−𝜌𝜋

𝑧2 (6.1)

During the hobbing and grinding process, the mobile reference system of each

component rotates around its rotation axis Z according to an angle θ.

After expressing Rp and Rh in function of Rp0 and Rh0 respectively, the

goniometric relations have been defined to express Rh0 as a function of Rp0.

76

A generic point M is defined in Rh0 as follows:

𝑂ℎ𝑀ℎ = 𝑥ℎ𝒏𝑥ℎ0 + 𝑦ℎ𝒏𝑦ℎ0 + 𝑧ℎ𝒏𝑧ℎ0 (6.2)

The same point M with respect to Rp0 is defined as follows:

𝑂𝑝𝑀ℎ = 𝑂𝑝𝑂ℎ + 𝑂ℎ𝑀ℎ

(6.3)

Where:

𝑂𝑝𝑂ℎ = 𝐶 𝒏𝑥𝑝0

𝒏𝑥ℎ0 = −𝒏𝑥𝑝0

𝒏𝑦ℎ0 = −𝑐𝑜𝑠(𝑆)𝒏𝑦𝑝0 + 𝑠𝑖𝑛(𝑆)𝒏𝑧𝑝0

𝒏𝑧ℎ0 = 𝑠𝑖𝑛(𝑆)𝒏𝑦𝑝0 + 𝑐𝑜𝑠(𝑆)𝒏𝑧𝑝0

Through the relationships listed above, it is possible to express Rh0 in Rpo.

Fig.6.6 shows the engagement between pinion and hob.

Fig.6.6 Pinion-Hob Engagement

77

6.4 Contact points during grinding process

The code to modelling the hobbing and grinding process could be considered the

same. This is possible because both processes work following similar dynamics.

The main difference between the hobbing and grinding process are:

- Hobbing process: non-continuous process

- Grinding process: continuous process

The procedure to obtain the contact points between pinion and hob depends on

the manufacturing process.

The kinematic data of the machine have been defined as follows:

Ωp=1 rpm

fh=0.2 mm/turn

The matrices have been defined to save the contact points.

The feed speed of the hob Vh is related to the angular speed of the pinion, and

is defined as follows:

𝑉ℎ =Ω𝑝

2𝜋 𝑓ℎ (6.4)

Between Ωp and Ωh there is a relationship depending on the type of cutting

motion direction. The relation is14:

78

Ωℎ = −𝑧1𝑧2[Ω𝑝 +

𝑡𝑎𝑛(𝛽1𝑏)

𝑟1𝑏(𝑠𝑖𝑛(𝑆 − 𝛽𝑎)

𝑠𝑖𝑛(𝑆)−𝑠𝑖𝑛(𝛽2)𝑠𝑖𝑛(𝛽𝑎)

𝑠𝑖𝑛(𝛽1)𝑠𝑖𝑛(𝑆)) ∙ 𝑉ℎ] (6.5)

In our case, the feed is axial.

The manufacturing process is not instantaneous but last for a certain time, for

this reason a vector of time tt has been defined. To define the vector tt it is

necessary to start from the definition of angular position of the pinion at the time

θpt14:

𝜃𝑝𝑡 = 𝜃𝑝 +𝑡𝑎𝑛(𝛽1𝑏)

𝑟1𝑏𝑧𝑝𝑡 + (𝑖 − 1)

2𝜋

𝑧1 (6.6)

With:

𝜃𝑝 = Ω𝑝 ∙ 𝑡 (6.7)

𝑧𝑝𝑡 =𝑠𝑖𝑛(𝑆 − 𝛽𝑎)

𝑠𝑖𝑛(𝑆)∆𝑉𝑡 (6.8)

∆𝑉𝑡 = 𝑉ℎ ∙ 𝑡 (6.9)

ΔVt represents the movement of the cutter along the feed direction, in the case

of axial feed, ΔVt will be equal to zpt. The rotation of the pinion is reversed with

respect to the convention initially applied, so the index the factor (i-1) will have

a negative sign.

79

The index i does not represent the tooth-i th but represents the transition from

one tooth to the next during the process, therefore the ii-th index does not

correspond to the real tooth ii-th but the relation between i and ii is:

ii=1 → i=1

ii=2 → i=z1

ii=3 → i=z1-1

It is possible to rewrite θpt as follows:

𝜃𝑝𝑡 = Ω𝑝 ∙ 𝑡 +𝑡𝑎𝑛(𝛽1𝑏)

𝑟1𝑏

Ω𝑝𝑓ℎ

2𝜋∙ 𝑡 − (𝑖𝑖 − 1)

2𝜋

𝑧1 (6.10)

It is possible to obtain 𝑡: 𝜃𝑝𝑡(𝑖𝑖, 𝑧𝑝𝑡, 𝑡) = 0 as follows:

𝑡 =(𝑖𝑖 − 1)

Ω𝑝

2𝜋

𝑧1(

1

1 +𝑡𝑎𝑛(𝛽1𝑏)𝑟1𝑏

𝑓ℎ2𝜋

) (6.11)

The term highlighted is a term << 1, so we can use the first-degree development

of Taylor and then get:

𝑡 =(𝑖𝑖 − 1)

Ω𝑝𝑧1∙ (2𝜋 −

𝑡𝑎𝑛(𝛽1𝑏)

𝑟1𝑏𝑓ℎ) (6.12)

80

In similar way, the angular position of the hob over time is defined as follows:

𝜃ℎ𝑡 = 𝜃ℎ +𝑡𝑎𝑛(𝛽2𝑏)

𝑟2𝑏𝑧ℎ𝑡 + (𝑖ℎ − 1)

2𝜋

𝑧2 (6.13)

With:

𝜃ℎ = Ωℎ ∙ 𝑡 + 𝜃ℎ0 (6.14)

𝑧ℎ𝑡 = −𝑠𝑖𝑛(𝛽𝑎)

𝑠𝑖𝑛(𝑆)∆𝑉𝑡 (6.15)

∆𝑉𝑡 = 𝑉ℎ ∙ 𝑡 (6.16)

Subsequently, a verification is performed taking into consideration the relation

reported13:

𝑟1𝑐𝑜𝑠(𝛽1) ∙ 𝜃𝑝𝑡 + 𝑟2𝑐𝑜𝑠(𝛽2) ∙ 𝜃ℎ𝑡 +1

2(𝑒𝑛1 + 𝑒𝑛2) = ∆∁ ∙ 𝑡𝑎𝑛(𝛼𝑛) (6.17)

Once the angular positions of pinion and hob have been defined, the coordinates

of the contact points on the surface of pinion and hob have been determined.

The following discussion refers to the pinion, but similarly has been done for the

hob, as reported in the code. We represent with zm the coordinate of the point of

contact on the surface of the pinion, and it is defined as follows:

𝑧𝑚 = 𝑧𝑚𝑡 + 𝑧𝑝𝑡 (6.18)

81

Zpt has been defined previously, such as the hobbing or grinding coordinate on

the pinion profile at time t. Zmt represents the coordinate of the point of contact

M starting from point C1 belonging to the axis of rotation of the pinion, and it is

defined as follows13:

𝑧𝑚𝑡 =1

𝑡𝑎𝑛(𝛽𝑏1)(𝑠1 − 𝑟𝑏1 ∙ (𝜃𝑝𝑡 + 𝑤1

𝜋

2𝑧1)) (6.19)

The term w1 represents a scaling factor of the theoretical tooth thickness. In the

case of the hob, the thickness scale factor is unitary. To define zmt it is therefore

necessary to define s1. S1 is the distance between the point of contact M and the

point P belonging to the line of action as shown in Fig.6.7.

S1 is defined as follows13:

𝑠1𝑐𝑜𝑠(𝛽𝑏1)

= 𝑐𝑜𝑠(𝛼𝑛) (∆∁1

𝑡𝑎𝑛(𝛼𝑛)+1

2𝑒𝑛1 + 𝑟1𝑐𝑜𝑠(𝛽1)𝜃𝑝𝑡) (6.20)

In similar way, s2 is defined for the hob.

82

Fig.6.7 Point of contact13

S1 can be expressed as a function of the apparent pressure angles of the contact

point and pitch radius, as follows14:

𝑠1 = 𝑟𝑏1(𝑡𝑎𝑛(𝛼𝑡𝑚1) − 𝑡𝑎𝑛(𝛼𝑡1)) (6.21)

Subsequently, a further check has been performed to ensure consistency between

the differents mathematical relationships.

83

To have a correct engagement and correct contact points it is necessary to verify

this relation14:

𝑟𝑏1𝑐𝑜𝑠(𝛽𝑏1)

(𝑡𝑎𝑛(𝛼𝑡𝑚1) − 𝑡𝑎𝑛(𝛼𝑡1)) +𝑟𝑏2

𝑐𝑜𝑠(𝛽𝑏2)(𝑡𝑎𝑛(𝛼𝑡𝑚2) − 𝑡𝑎𝑛(𝛼𝑡2)) =

∆∁

𝑠𝑖𝑛(𝛼𝑛)(6.22)

To obtain the xm and ym coordinates of the contact point M, a vector rm has

been defined. It represents the radius with center in C1 on which the different

contact points are positioned, as shown in Fig.6.7. On the apparent plane of the

tooth, a point of contact is also a point of cutting or grinding.

In the code θm it has a negative sign, because the sense of rotation of the pinion

is reverse. A verify has been done to ensure rm between the tip radius and root

radius. In a similar way, the coordinates of the contact points belonging to the

hob have been obtained.

The matrices that have been obtained [Mpx, Mpy, Mpz] and [Moux, Mouy, Mouz]

are respectively the matrices of the contact points during the grinding process

belonging to the surface of the pinion and the hob.

Fig.6.8 shows the contact points on each tooth of the pinion and on the hob.

84

Fig.6.8 Grinding contact points

6.5 Contact points during the hobbing process

To obtain contact points during the hobbing process we must select the contact

points that we have previously obtained. The selection criterion is based on the

flutes: a point of contact is also a cutting point if this point belongs to a flute of

the hob. As mentioned previously, the hobbing process is not a continuous

85

process but there are stops when moving from one flute to the next, so the cutting

points are exclusively the points belonging to the flute.

To select the points of cutting, it depends on the type of flute Fig.6.9:

- Helical flute

- Straight flute

The flute developed along the perpendicular of the hob fillet are called helical

flute. The flute developed along the axis of rotation of the hob are called straight

flute.

Fig.6.9 Straight and helical flutes30

A straight flute does not allow to obtain an optimal hobbing process as it could

be with a helical flute hob, but the process of sharpening on straight flute hob is

certainly less complicated than a helical flute hob.

86

We have a hob with straight flutes, then we consider the relation as follows:

A point M, belonging to the hob whose cylindrical coordinates are rmh, θmh, zmh,

is a cutting point if its cylindrical coordinates satisfy the equation of belonging

to flute k-th 14:

𝜃𝑚ℎ = 𝜃𝑘0 + (𝑘 − 1) ∙2𝜋

𝑁𝑏𝑔 (6.22)

The matrices that have been obtained [Mp2x, Mp2y, Mp2z] and [Mou2x, Mou2y, Mou2z]

are respectively the matrices of the contact points during the hobbing process

belonging to the surface of the pinion and the hob. Fig.6.10 shows the contact

points on each profile of pinion and hob.

Fig.6.10 Hobbing contact points

87

7

Conclusion

7.1 Project development

The project was based on an accurate theoretical study on the phenomenon of

sound in speed reducers, which is a current study and takes in many companies

in the automotive sector. The main causes that have been attributed to the sound

phenomenon are manufacturing defects that inevitably lead to geometric defects

of components that are part of a gear. For this reason, it was decided to analyze

the hobbing process and the grinding process to which a tooth wheel is usually

subjected.

In order to analyze the hobbing and grinding process, a detailed study was

carried out on the hobbing kinematics of a pinion through a hob. The data

characterizing the pinion and the hob are real. In order to model the hobbing

process that generates the helical wheel, the kinematic relations between the

hob and the pinion have been studied during the engagement.

A matlab code was implemented to generate the pinion and the hob in three

dimensions, then the engagement conditions were defined, and the theoretical

contact points were then obtained during the hobbing and grinding process on

88

the surface of the pinion and on the surface of the hob. To draw the gear models

by MATLAB codes is difficult at the beginning. Firstly, all the functions and

relationships of gear curves should be found out from mathematic and geometric

point of view. Secondly, all the curves of gear models are made up of a series of

nodes in matrix form, so how to handle with matrix operation like matrix

addition, matrix rotation should be learnt. Thirdly, proper MATLAB

programming languages and functions should be used, such as ‘for’ loop codes,

‘flip’ function, ‘cat’ function, and ‘mesh’ function etc.

7.2 Forthcoming objectives

Now that the modelling code of the hobbing and grinding process has been

written, it will be possible to subsequently extrapolate the theoretical contact

points and compare these with the real contact points and note any differences.

To understand the possible difference between the theoretical and the actual

simulation, it will be necessary to introduce some tooth profile modification on

the gear models, based on the theoretical study carried out in the project on the

different types of defects. This is possible because MATLAB can import data

from .mat file.

Then, every time doing the tooth profile modification, just load the tooth profile

modification code files to the tooth model files, the model with tooth profile

modification can be obtained. By MATLAB GUI it is possible to provide a more

straight and simpler way for users to obtain gear models and to do tooth profile

89

modification, because users just must input and click buttons in the interface

without knowing the programming codes.

Subsequently it will be necessary to obtain the meshing frequencies between the

theoretical engagement and the real engagement and to understand which of the

different defects introduced gives a theoretical result close to the real one.

90

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